An Energy-Based Conservative-Dissipative Latent Neural Evolution Operator for Magnetization Dynamics

1 Introduction

• The motivation is that LLG dynamics combine precession and damping, while the effective field is derived from the Gibbs free-energy functional.The paper asks whether this scalar-driving property can be useful for machine learning.Source: §1 · pp. 1–2
• Repeated micromagnetic solves can bottleneck parameter studies, optimization, and inverse design because of problem stiffness and the cost of numerical integration.Nonlinear-manifold autoencoders offer reduced coordinates when trajectories occupy a low-dimensional manifold despite high-dimensional spatial discretization.Source: §1 · p. 2
• An unrestricted neural ODE distinguishes neither circulation from dissipation nor a scalar constrained to decrease, and short-window agreement alone may not prevent accumulated rollout error.Neural ODEs parameterize continuous-time vector fields and permit evaluation at requested times.Source: §1 · pp. 2–3
• A structured latent field uses the learned scalar's gradient with antisymmetric and symmetric positive-semidefinite operators to represent level-set motion and decreasing dynamics.The learned scalar is defined in nonunique latent coordinates and is not identified with Gibbs free energy.Source: §1 · p. 3
• The approach jointly trains a spatial latent-tensor encoder–decoder and conservative–dissipative dynamics using latent and decoded-rollout losses, without physical-energy labels or derivative supervision.The listed contributions also include an autonomous continuous-time dissipation identity and comparisons across three latent-energy models.Source: §1 · p. 3

2.1 Observed dynamics and magnetization states

• The objective is a reduced flow for trajectory families approximated by a lower-dimensional smooth manifold, using snapshot sequences without requiring derivatives or scalar-energy labels.The formulation does not aim to reconstruct a full governing equation throughout the observation space.Source: §2.1 · p. 3
• With N finite-difference cells, magnetization is represented by N three-component vectors, each constrained to unit length, so the physical state space is a product of spheres.Source: §2.1 · p. 3

2.2 Nonlocal interactions and representational scope

• The nonlocal demagnetization field depends on magnetization throughout the sample and is a major computational burden in micromagnetic simulation.Traditional solvers must compute it at every step, which can be especially costly for small explicit-solver steps.Source: §2.2 · pp. 3–4
• The latent model avoids evaluating the demagnetization field at every ODE step and may represent its effects implicitly from data.It learns an internal approximation to local and nonlocal interactions.Source: §2.2 · p. 4

2.3 Energy decay of the reference dynamics

• For a constant applied field, the reference LLG dynamics have non-increasing total Gibbs free energy under the stated unit-magnetization and parameter conditions.The dissipation rate is proportional to α and vanishes at equilibrium.Source: §2.3 · p. 4
• Precession contributes zero to Gibbs-energy change, while damping yields a nonnegative dissipation rate that vanishes when the effective field is parallel to magnetization.This equilibrium condition follows from the stated calculation.Source: §2.3 · p. 4
• Only total energy is guaranteed to be monotone; individual contributions can vary, with exchange energy typically increasing during reversal-structure formation.The latent model imposes an analogous one-scalar decrease for a constant conditional parameter.Source: §2.3 · p. 4

3 Model architecture

• The architecture consists of an autoencoder and a latent vector-field model acting on a reduced space Z = R^d.The encoder, decoder, energy model, and operator parameters are denoted θ, ξ, η, and ω, respectively.Source: §3 · p. 4
• Inference encodes the initial observation once, evolves its latent representation to requested times, and decodes the resulting states into observations.Physical parameters condition the scalar potential and therefore the latent vector field.Source: §3 · p. 4, Figure 1 · p. 5

4.1 Scalar potential and gradient force

• The learned force is the gradient of a differentiable scalar potential with respect to latent state z, holding trajectory parameter p fixed.A deep scalar permits its gradient and Hessian to vary with state, whereas an affine scalar yields constant velocity in an autonomous model.Source: §4.1 · p. 5

4.2 Conservative–dissipative split

• For fixed p, the antisymmetric operator generates motion along level sets of the learned scalar, while the positive-semidefinite operator generates descent and yields an exact continuous-time decrease for autonomous latent flow.This decrease does not identify the learned scalar with physical energy; monotonicity is guaranteed when p is constant, or when parameter variation contributes a nonpositive term.Source: §4.2 · pp. 5–6

4.3 Spectral properties of the operators

• The eigenvalues of J are purely imaginary or zero, while those of R are real and nonnegative.The spectra can help analyze latent directions and estimate rank for a potential low-rank ansatz.Source: §4.3 · pp. 6–7

4.4 Conservative motion, dissipation, and equilibria

• Every critical point is an equilibrium, but the converse need not hold; zero dissipation can coexist with motion under J.Dissipation alone does not establish asymptotic stability, which also depends on local energy shape and operator nullspaces.Source: §4.4 · p. 7

4.5 Nonlinear expressivity and local dynamics

• The deep–quadratic energy combines an affine contribution from its quadratic term with a nonlinear correction from its deep term, retaining the conservative–dissipative structure.A deep energy allows the gradient and Hessian to vary with state, unlike the affine vector field produced by a quadratic energy.Source: §4.1 · p. 5, §4.5 · pp. 7–8

4.6 Potential offset and scale

• Trajectory supervision cannot determine absolute energy or operator scale, and physical interpretation would require additional calibration or physical-energy supervision.The energy offset is also unidentifiable because adding a parameter-dependent scalar does not change the gradient.Source: §4.6 · p. 8
• Energy supervision may increase latent-dynamics stiffness, and even with such supervision, an accurate effective-field approximation requires snapshots sampled at a sufficient temporal rate.Whether increased temporal resolution itself increases stiffness remains open.Source: §4.6 · p. 8

5.1 Short windows extracted from long trajectories

• Complete trajectories are split into contiguous k-snapshot windows after train, validation, and test assignment, preventing snapshots from one trajectory crossing partitions.Windows may overlap, and each extracted window is treated as an independent training sample.Source: §5.1 · p. 8

5.2 Latent trajectory loss

• The latent trajectory loss compares ODE-integrated states with encoder-derived references, encouraging coordinates whose evolution fits the structured latent ODE.Because the encoder also determines the targets, this loss depends on latent coordinates and can change under their rescaling.Source: §5.1 · pp. 8–9, §5.2 · p. 9

5.3 Decoded rollout loss and joint objective

• The decoded rollout loss compares integrated latent states after decoding with observations, while the joint objective also retains the latent trajectory loss.It trains the decoder on states reached by numerical integration and includes exact reconstruction of the initial state, which notably improves training; all parameters are optimized jointly.Source: §5.3 · p. 9

6.1 Neural ODE viewpoint

• Unlike fixed-depth networks, a neural ODE uses a numerical solver to handle requested output times, and the structured latent field imposes a more restrictive inductive bias than an unrestricted neural ODE.Training differentiates the loss through the numerical solution map.Source: §6.1 · pp. 9–10

6.2 Reverse-over-reverse differentiation

• Reverse-over-reverse differentiation computes Hessian–vector and mixed products through the scalar gradient without constructing dense derivative matrices.The outer pass differentiates through the inner reverse pass and numerical solve; adaptive solvers can yield different gradients under continuous adjoints versus direct discrete differentiation at finite tolerance.Source: §6.2 · p. 10

6.3 Continuous and discrete dissipation

• The monotonicity identity applies to the exact continuous solution, not necessarily to every accepted step or saved output of a finite-tolerance integrator.Small numerical increases do not violate the analytic identity when the operators satisfy the stated conditions; solver choices and tolerances are evaluation choices.Source: §6.3 · p. 10

6.4 Computational scaling

• Dense operators require O(d^2) storage and approximately Cg + O(d^2) work per field evaluation; rank-r factorizations could reduce these to O(dr) when r ≪ d.The rank restrictions may reduce expressivity and are not studied; retaining spatial dimensions changes the cost to Cg + O(|S|d^2), and speed claims require accounting for encoding, integration, decoding, and accuracy.Source: §6.4 · pp. 10–11

• The learned antisymmetric channel operator preserves the scalar algebraically, while the positive-semidefinite operator decreases it, paralleling features of Hamiltonian and gradient systems.The latent coordinates are not assumed canonical, and the channel operator is learned.Source: §7 · p. 11
• Unlike complete GENERIC formulations, the one-scalar construction does not distinguish energy and entropy generators or impose their additional degeneracy conditions.Compared with an unrestricted latent neural ODE, it uses a smaller hypothesis class; whether that trade-off is beneficial requires empirical comparison.Source: §7 · p. 11

8 Numerical results

• The datasets cover 200 field amplitudes per direction, with 199 non-reference trajectories split into 139 training, 20 validation, and 40 test trajectories.The field 1 and field 2 amplitude ranges are 20.00–29.95 mT and 30.00–39.95 mT, respectively; trajectories span 0–1 ns on a 100 × 25 × 1 grid.Source: §8 · p. 11

8.1 Implementation and optimization

• Six models cross dissipative-only and antisymmetric–dissipative vector fields with quadratic, deep, and additive deep–quadratic latent energies.The deep–quadratic energy combines an affine contribution with a nonlinear correction; spatial convolutions provide cross-site coupling in latent dynamics.Source: §8.1 · p. 12
• Configurations and training choices were fixed rather than systematically tuned, so the reported results are not estimates of best attainable performance.The study focuses on one grid and two field directions and does not quantify optimization variability.Source: §8 · p. 11, §8.1 · p. 13

8.2 Joint vector-field and energy comparison

• 0.0459 for field 1 and 0.2929 for field 2 are the RMSEs of the antisymmetric–dissipative deep–quadratic model, the lowest in this six-model, single-seed comparison.The tables pool 40 held-out trajectories per field.Source: §8.2 · p. 13, Table 2 · p. 13, Table 3 · p. 14
• Small windowed-test errors do not guarantee accurate uninterrupted rollouts: dissipative-only models have much larger rollout errors and fail to reproduce the full trajectories in both comparisons.The authors conclude that dissipation alone appears too restrictive for accurately learning these latent dynamics.Source: §8.2 · p. 14

8.3 Reference trajectories, learned energy, and operator diagnostics

• The learned latent-energy traces were non-increasing, but latent energy did not qualitatively coincide with Gibbs energy; field 1 was captured well, while field 2 had substantially larger error.Operator spectra showed a broader imaginary-axis spread for the antisymmetric operator than the dissipative eigenvalues clustered near zero.Source: §8.3 · pp. 14–15

8.4 Extrapolation beyond the training horizon

• At 2 ns, the deep–quadratic model stayed close to the reference for field 1 while the deep model departed and relaxed toward a spurious state; deep–quadratic also performed better for field 2, less clearly.The 1 ns marker in the paired rollouts separates the training interval from extrapolation.Source: §8.4 · p. 15, Figure 12 · p. 20, Figure 13 · p. 21

9 Limitations and prospective applications

• The experiments use one grid and two field directions, omit hyperparameter tuning, and do not quantify optimization variability.The authors caution that the results are not estimates of best attainable performance.Source: §9 · p. 22
• The dissipation identity alone does not establish trajectory accuracy, physical-energy recovery, stability, or robustness outside the sampled family.Source: §9 · p. 22
• Larger conditional parameter spaces are proposed for future study, including field magnitude and angle, replay-buffer transitions, and tests of spatial generalization across input shapes.Source: §9 · p. 22
• A compact latent rollout may suit integrated sensing only if task-specific accuracy, latency, and robustness are demonstrated.When intermediate fields are unnecessary, the authors propose decoding only a terminal state or using a lightweight latent-to-observable head.Source: §9 · p. 22

10 Conclusion

• The reduced-order model couples an autoencoder to an energy-gradient ODE whose autonomous flow decreases the learned latent scalar while antisymmetric dynamics permit motion along its contour lines.Source: §10 · p. 22
• Results provide empirical evidence that the antisymmetric operator is required for accurate conditioned dynamics, while deep–quadratic energy supports realistic field-1 generalization past the training horizon.The reported generalization is in paired post-hoc 2 ns rollouts.Source: §10 · p. 22
• The source code used to produce the results is publicly available on Zenodo.Source: §10 · p. 22